On Instability of the Rayleigh–Bénard Problem without Thermal Diffusion in a Bounded Domain under L1-Norm

Pan Zhang,Mengmeng Liu and Fangying Song

School of Mathematics and Statistics,Fuzhou University,Fuzhou,Fujian 350108,China

Abstract.We investigate the thermal instability of a three-dimensional Rayleigh–Bénard(RB for short)problem without thermal diffusion in a bounded domain.First we construct unstable solutions in exponential growth modes for the linear RB problem.Then we derive energy estimates for the nonlinear solutions by a method of a prior energy estimates,and establish a Gronwall-type energy inequality for the nonlinear solutions.Finally,we estimate for the error of L1-norm between the both solutions of the linear and nonlinear problems,and prove the existence of escape times of nonlinear solutions.Thus we get the instability of nonlinear solutions under L1-norm.

Key words: Rayleigh–Bénard problem,thermal instability,initial-boundary value problem.

1 Introduction

Thermal instability often arises when a fluid is heated from below.The phenomenon of thermal convection itself had been recognized by Rumford[24]and Thomson[25].However,the first quantitative experiment on thermal instability and the recognition of the role of viscosity in the phenomenon are due to Bénard [2].The Bénard convection can be modeled by the(nonlinear)compressible Navier–Stokes–Fourier(simplified by NSF) equations [11].Since the continuity equation in the NSF equations is hyperbolic,thus it is difficult to theoretically investigate the thermal convection.Later Rayleigh investigated thermal convection by using Boussinesq (approximation) equations,in which the density is considered as a constant in all the terms of the equations except for the gravity term that is assumed to vary linearly with the temperature [4].Compared with the compressible NSF model,the Boussinesq model is fully parabolic due to the absence of the continuity equation.Thus,based on the linearized Boussinesq model,Rayleigh first theoretically provided the instability criterion for the occurrence of thermal convection [3,23].After Rayleigh’s pioneering work,the instability criterion had been further mathematically verified for the nonlinear Boussinesq model in the Hadamard sense by the energy method and the bootstrap instability method,see [13,20] for examples.At present,it has been also widely investigated how the thermal instability evolves under the effects of other physical factors,such as the elasticity [19],rotation [7,10],the magnetic field [6,8,9],surface tension [22] and so on.Recently Ma and Wang also established mathematical theory of attractor bifurcation for two-dimensional Boussinesq model[21].However,the corresponding three-dimensional case is still an open problem.

It is physically well-known that the system of nonlinear Boussinesq equations is always unstable,if the thermal diffusion is absent.However there is not any available mathematical proof for this physical assertion.In this paper,we try to mathematically prove this assertion.To begin with,we shall introduce the three-dimensional(3D for short) Rayleigh–Bénard (RB for short) equations without thermal diffusion in a bounded domain Ω:

Next we further explain the notations in the equations above.

The unknownsvv(x,t),ΘΘ(x,t) andpp(x,t) denote velocity,temperature and pressure of an incompressible fluid,resp..The parametersρ,α,μ＞0 andg ＞0 denote the density constant at some properly chosen temperature parameter Θb,the coefficient of volume expansion,shear viscosity coefficient,and the gravitational constant,respectively.gρα(Θ-Θb)e3stands for the buoyancy,-ρge3for the gravitational force,wheree3(0,0,1)Tand T denotes the transposition.

The rest state of the system(1.1)can be given byrB:(0,)with an associated pressure profile ˉp,where the temperature profile ˉΘ and ˉpdepend onx3only,and satisfy the equilibrium state equations:

For the sake of simplicity,we only consider the case:

where?＞0 is a constant of adverse temperature gradient.

Denoting the perturbation of (v,Θ) aroundrBby

then,(v,θ,q) satisfies the perturbation equations in Ω:

We call (1.4) the perturbed RB equations.For the system (1.4),we impose the following initial-boundary value conditions:

Moreover,the compatibility condition divv00 should be satisfied.Next we state our main result.

where Tmaxdenotes the maximal time of existence of the solution(θ,v).

Remark 1.1.Here in what follows,we have used the simplified notations of Sobolev’s spaces: for 1≤p≤∞and the non-negative integerk,

In addition,for the sake of simplicity,we will use the following simplified notations:

which means thata ≤cbfor some constantc,where the positive constantcmay depend on the domain occupied by the fluids and other known physical parameters such asg,?,αandμ,and vary from line to line.

The proof of Theorem 1.1 is based on a bootstrap instability method.The bootstrap instability method has its origin in [14,15].Later,many authors further developed various versions of bootstrap approaches,see [5,12,26] for examples.In this paper,we adopt a bootstrap instability method in [18] to prove Theorem 1.1.Firstly,we shall construct unstable solutions to the linearized RB problem (2.1) by using a variational method in [16] due to the presence of viscosity,see Proposition 2.1 in Section 2.Secondly,we further establish a Gronwall-type energy inequality for the non-linear solutions of the RB problem (1.4)–(1.5b),see Proposition 3.1 in Section 3.Thirdly,we deduce the error estimates ofL1-norm between the both solutions of the linearized and nonlinear problems,see Proposition 4.1 in Section 4.Finally,we prove the existence of escape times and thus obtain Theorem 1.1 in last section.

2 Linear instability

Using the variational method of PDE in [16] and an regularity theory of Stokes equations,we can obtain the following linear instability result of the RB problem.

Under the assumptions of Theorem1.1,the equilibrium state rB of RB problem is linearly unstable,that is,there is an unstable solution in the form

to the following linearized system(1.4)around the equilibrium state rB:

wheresolves the following boundary problem

withΛ＞0being a constant satisfying

In addition,

wheredenotes the i-th component of v0for1≤i≤3.

Proof.To begin with,we assume a growing mode ansatz of solutions,i.e.,for some Λ＞0,

Putting the above ansatz into (2.1),we get

Multiplying (2.5a) and (2.5b) by/?and/gαinL2,respectively,adding the two resulting identities together,and then using the integral by parts,we get

Next we find a Λ by maximizing

Noting that

thus,by Young’s and Poincaré’s inequalities,we see that

Moreover we easily see that there is (v,θ),such that

which yields

For anyR,2and,we take

then (2.7) implies

If we set

thenI(τ)1(R),I(τ)≤0 for allR andI(0)0.ThusI′(0)0.By a direct computation,we have

which implies that

Applying the classical regularity theory of the Stokes equations to(2.10),there exists a function1such that

Finally,noting that

In addition,we can further exploit an energy method to show thateΛtis indeed the sharp growth rate for (θ,v) inH2-norm,i.e.,we have the following conclusion.

Proposition 2.2.Let(θ,v,q)be a solution of the following linearized problem

Then the following estimates hold for any t≥0,

whereΛis constructed by(2.3).

Proof.Let (θ,v,q) be the solution of (2.11).Applyingto (2.11a) and (2.11b),we have,fori0 and 1,

Multiplying (2.13a) and (2.13b) byandinL2,respectively,adding the resulting equalities together,and then using the integral by parts and the facts divvt0 andvt|?Ω0,one has

Thanks to (2.3),we have

Consequently,one has

which,together with Gronwall’s inequality,yields

Therefore,we arrive at that

Multiplying (2.11a) by?qinL2,we get

which implies that

In particular,

Thus we further see from (2.11a) and (2.11b) that

Putting the above estimate into (2.15) yields (2.12a).

Now we turn to the proof of (2.12b).We easily deduce from (2.11) that

Using Poincáre’s and Young’s inequalities,we further have

which,together with (2.15) withi0 and (2.17),yields

Applying the classical regularity theory on the Stokes equations to (2.11a),we get

Thus we derive from (2.15) withi0,(2.19) and (2.20) withj0 that

Letαsatisfy|α|≤2.Applying?αto (2.11b) and then multiplying the resulting equation by?αθinL2,we get

By H¨older’s inequality,we further have

which,together with (2.21),yields

Consequently,we derive (2.12b) from (2.20) withj1,(2.21) and (2.24).This completes the proof of Proposition 2.2.

3 Gronwall energy inequality

Now we turn to the derivation of Gronwall-type energy inequality for the nonlinear solution of the RB problem (1.4)–(1.5b).To this end,let (θ,v) be a solution of the RB problem,such that

whereδis sufficiently small.It should be noted that the smallness depends on the domain and other konwn physical parameters in the RB problem,and will be repeatedly used in what follows.To begin with,let us first recall some basic estimates.

Lemma 3.1.(1) Embedding inequalities(see [1,Theorem4.12]):

(2) Product estimates in Sobolev spaces:

which can be easily verified by H¨older’s inequality and the embedding inequality(3.2a)–(3.2b).

(3) Interpolation inequality in Hj(see [1,Theorem5.2]):

where the constant C(ε)depends on the domain and ε,and Young’s inequality has been used in the last inequality in(3.4).

Next we establish some estimates for the solution (θ,v).

Lemma 3.2.Under the assumption(3.1)with sufficiently small δ,we have

Proof.Letαsatisfy|α|≤2.Applying?αto (1.4b) and multiplying the resulting equation by?αθinL2,we get

Using the product estimate (3.3) and the integral by parts,we can derive (3.5a)from the above identity.In addition,we easily derive from (1.4b) that

which,together with (3.1) and the product estimate (3.3),implies (3.5b).

Lemma 3.3.Under the assumption(3.1)with sufficiently small δ,the following estimates hold

Proof.Next we derive the five estimates (3.8a)–(3.8d) in sequence.

(1).By (1.4a),(3.1) and (3.3),we have

Multiplying (1.4a) by?qinL2,then using the integral by parts and the facts divvt0 andvt|?Ω0,we get

which implies that

Putting the above estimate into (3.9) yields (3.8a).

(2).Letk0 and 1.Applying the classical regularity theory of the Stokes equations to (1.4a),we have

Using (3.1) and product estimate (3.3),we get

which yields (3.8b).

(3).Multiplying (1.4a)byvinL2,and then using integrating by parts,H¨older’s inequality and Poincáre’s inequality,we conclude that

which yields (3.8c).

(4).Applying?tto(1.4a)and multiplying the resulting equation byvtinL2,we deduce that

Using Young’s and Poincáre’s inequalities,product estimate,(3.1) and (3.5b),we get (3.8d) for sufficiently smallδ.This completes the proof of Lemma 3.3.

Lemma 3.4.Under the assumption(3.1),we have

Here and in what follows

Proof.By (3.8a) and (3.8b),we see that

which yields (3.15).

Thanks to the estimates of (θ,v) in Lemmas 3.2–3.4,we are in the position to derivea priorGronwall-type energy inequality for the nonlinear solutions of (1.4)–(1.5b).

Proposition 3.1.There exists a constant δ1(0,1),and C ＞0,such that,for any δ≤δ1,if the solution(θ,v)of RB problem satisfies(3.1)(0,T),the solution(θ,v)satisfies the Gronwall-type energy inequality

Proof.We can derive from (3.5a),(3.8c) and (3.8d) that

where we have defined that

Integrating (3.18) over (0,t),and then using (3.8b) and (3.15),we have

Consequently,by (3.1) and (3.4),we immediately derive (3.17) from (3.19).

Using the standard iteration scheme as in [17,Proposition 3.1],we can easily establish the existence of unique solution for the RB problem(1.4)–(1.5b).Moreover,the solution enjoys the Gronwall-type energy inequality.More precisely,we have the following conclusion:

Proposition 3.2.Let initial data(θ0,v0)22(0,1),such that,if(θ0,v0)satisfying

then there exists a local existence time Tmax＞0,depending on δ2,the domain and the other known physical parameters,and a unique strong solution(θ,v,q)([0,Tmax),H21)to the RB problem(1.4)–(1.5b).Moveover,if E ≤δ1for some interval IT,which belongs to ,then the solution further satisfies the Gronwall-type energy inequality(3.17).

4 Nonlinear instability

Now we are in a position to prove Theorem 1.1 by adopting the basic idea in [16].First,in view of Proposition 2.1,we can construct a linear solution

We denoteTmin:min{Tδ,T*,T**}.By the definition ofT**,we can deduce from the estimate (3.17) that,for allt＜Tmin,

Applying Gronwall’s inequality to the above estimate,we arrive at,for some constantC2,

Let

where we have defined that

Proposition 4.1.There exists a constant C3such that,for any δ(0,1),

where C3is independent of Tmin.

Proof.Let us recall the fact that (θd,vd) satisfies the following non-homogeneous equations:

with initial data

Multiplying (4.8a)and(4.8b)byθd/?andvd/gαinL2,respectively,adding the resulting equalities and then using (4.6),we get,for anyt＜Tmin,

In addition,similarly to (2.14),we have

Thus,putting the above estimate into (4.9),we get

Applying Gronwall’s inequality to the above inequality yields that

for anyt＜Tmin.Noting thatL21,then we derive(4.7)from(4.11).We complete the proof of Proposition 4.1.

Let

It is easy to see thatm0＞0 by (4.2a).Now,we assert that

which can be proved by contradiction as follows:

(1).IfTminT*,thenT*≤Tδ ＜∞.MoreoverT*＜Tmaxby Proposition 3.2.Noting that0/4,we can deduce from (4.3a) and (4.6) we get

which contracts (4.4a).HenceTminT*.

(2).IfTminT**,thenT**＜T*≤Tmax.Moreover,making use of (4.1),(4.3a)and (4.11),we see that

which also contradicts(4.4b).Therefore,Tmin.We immediately see that(4.13)holds.This completes the claim of (4.13).

SinceTδ ＜T*≤Tmax,we can use (4.7) and (4.12b) to deduce that

Similarly,we also have

wheredenote thei-th component ofvδ(Tδ)for 1≤i≤3.This completes the proof of the Theorem 1.1 by definingε:m0ε0/2.

Acknowledgements

The authors would like to thank the anonymous referee for invaluable suggestions,which improve the presentation of this paper.This work was supported by the NSF of China (Grant No.11901100),the Natural Science Foundation of Fujian Province of China (Grant No.2020J02001) and Funds of Education Department of Fujian Province (Grant No.510881/GXRC-20046).